| Step | Hyp | Ref
| Expression |
| 1 | | nfv 1914 |
. . . . 5
⊢
Ⅎ𝑘𝜑 |
| 2 | | nfv 1914 |
. . . . . 6
⊢
Ⅎ𝑘 𝑗 ∈ 𝑍 |
| 3 | | nfra1 3284 |
. . . . . 6
⊢
Ⅎ𝑘∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷) |
| 4 | 2, 3 | nfan 1899 |
. . . . 5
⊢
Ⅎ𝑘(𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) |
| 5 | 1, 4 | nfan 1899 |
. . . 4
⊢
Ⅎ𝑘(𝜑 ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷))) |
| 6 | | climxrrelem.z |
. . . . . . . . 9
⊢ 𝑍 =
(ℤ≥‘𝑀) |
| 7 | 6 | uztrn2 12897 |
. . . . . . . 8
⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → 𝑘 ∈ 𝑍) |
| 8 | 7 | adantll 714 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → 𝑘 ∈ 𝑍) |
| 9 | | climxrrelem.f |
. . . . . . . . 9
⊢ (𝜑 → 𝐹:𝑍⟶ℝ*) |
| 10 | 9 | fdmd 6746 |
. . . . . . . 8
⊢ (𝜑 → dom 𝐹 = 𝑍) |
| 11 | 10 | ad2antrr 726 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → dom 𝐹 = 𝑍) |
| 12 | 8, 11 | eleqtrrd 2844 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → 𝑘 ∈ dom 𝐹) |
| 13 | 12 | adantlrr 721 |
. . . . 5
⊢ (((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷))) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → 𝑘 ∈ dom 𝐹) |
| 14 | | simpll 767 |
. . . . . 6
⊢ (((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷))) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → 𝜑) |
| 15 | 8 | adantlrr 721 |
. . . . . 6
⊢ (((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷))) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → 𝑘 ∈ 𝑍) |
| 16 | | rspa 3248 |
. . . . . . . 8
⊢
((∀𝑘 ∈
(ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) |
| 17 | 16 | adantll 714 |
. . . . . . 7
⊢ (((𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) |
| 18 | 17 | adantll 714 |
. . . . . 6
⊢ (((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷))) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) |
| 19 | 9 | ffvelcdmda 7104 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈
ℝ*) |
| 20 | 19 | 3adant3 1133 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍 ∧ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) → (𝐹‘𝑘) ∈
ℝ*) |
| 21 | | simpll 767 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝐹‘𝑘) ∈ ℂ) ∧ (𝐹‘𝑘) = -∞) → 𝜑) |
| 22 | | simpr 484 |
. . . . . . . . . . . . . 14
⊢ (((𝐹‘𝑘) ∈ ℂ ∧ (𝐹‘𝑘) = -∞) → (𝐹‘𝑘) = -∞) |
| 23 | | simpl 482 |
. . . . . . . . . . . . . 14
⊢ (((𝐹‘𝑘) ∈ ℂ ∧ (𝐹‘𝑘) = -∞) → (𝐹‘𝑘) ∈ ℂ) |
| 24 | 22, 23 | eqeltrrd 2842 |
. . . . . . . . . . . . 13
⊢ (((𝐹‘𝑘) ∈ ℂ ∧ (𝐹‘𝑘) = -∞) → -∞ ∈
ℂ) |
| 25 | 24 | adantll 714 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝐹‘𝑘) ∈ ℂ) ∧ (𝐹‘𝑘) = -∞) → -∞ ∈
ℂ) |
| 26 | | climxrrelem.n |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ -∞ ∈ ℂ)
→ 𝐷 ≤
(abs‘(-∞ − 𝐴))) |
| 27 | 21, 25, 26 | syl2anc 584 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝐹‘𝑘) ∈ ℂ) ∧ (𝐹‘𝑘) = -∞) → 𝐷 ≤ (abs‘(-∞ − 𝐴))) |
| 28 | 27 | adantlrr 721 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) ∧ (𝐹‘𝑘) = -∞) → 𝐷 ≤ (abs‘(-∞ − 𝐴))) |
| 29 | | fvoveq1 7454 |
. . . . . . . . . . . . . . 15
⊢ ((𝐹‘𝑘) = -∞ → (abs‘((𝐹‘𝑘) − 𝐴)) = (abs‘(-∞ − 𝐴))) |
| 30 | 29 | adantl 481 |
. . . . . . . . . . . . . 14
⊢
(((abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷 ∧ (𝐹‘𝑘) = -∞) → (abs‘((𝐹‘𝑘) − 𝐴)) = (abs‘(-∞ − 𝐴))) |
| 31 | | simpl 482 |
. . . . . . . . . . . . . 14
⊢
(((abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷 ∧ (𝐹‘𝑘) = -∞) → (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷) |
| 32 | 30, 31 | eqbrtrrd 5167 |
. . . . . . . . . . . . 13
⊢
(((abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷 ∧ (𝐹‘𝑘) = -∞) → (abs‘(-∞
− 𝐴)) < 𝐷) |
| 33 | 32 | adantll 714 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷) ∧ (𝐹‘𝑘) = -∞) → (abs‘(-∞
− 𝐴)) < 𝐷) |
| 34 | 33 | adantlrl 720 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) ∧ (𝐹‘𝑘) = -∞) → (abs‘(-∞
− 𝐴)) < 𝐷) |
| 35 | | climxrrelem.c |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 𝐹 ⇝ 𝐴) |
| 36 | 6 | fvexi 6920 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ 𝑍 ∈ V |
| 37 | 36 | a1i 11 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → 𝑍 ∈ V) |
| 38 | 9, 37 | fexd 7247 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → 𝐹 ∈ V) |
| 39 | | eqidd 2738 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑘 ∈ ℤ) → (𝐹‘𝑘) = (𝐹‘𝑘)) |
| 40 | 38, 39 | clim 15530 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → (𝐹 ⇝ 𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+
∃𝑗 ∈ ℤ
∀𝑘 ∈
(ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥)))) |
| 41 | 35, 40 | mpbid 232 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+
∃𝑗 ∈ ℤ
∀𝑘 ∈
(ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥))) |
| 42 | 41 | simpld 494 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝐴 ∈ ℂ) |
| 43 | 42 | ad2antrr 726 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (𝐹‘𝑘) ∈ ℂ) ∧ (𝐹‘𝑘) = -∞) → 𝐴 ∈ ℂ) |
| 44 | 25, 43 | subcld 11620 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝐹‘𝑘) ∈ ℂ) ∧ (𝐹‘𝑘) = -∞) → (-∞ − 𝐴) ∈
ℂ) |
| 45 | 44 | abscld 15475 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝐹‘𝑘) ∈ ℂ) ∧ (𝐹‘𝑘) = -∞) → (abs‘(-∞
− 𝐴)) ∈
ℝ) |
| 46 | 45 | adantlrr 721 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) ∧ (𝐹‘𝑘) = -∞) → (abs‘(-∞
− 𝐴)) ∈
ℝ) |
| 47 | | climxrrelem.d |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝐷 ∈
ℝ+) |
| 48 | 47 | rpred 13077 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝐷 ∈ ℝ) |
| 49 | 48 | ad2antrr 726 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) ∧ (𝐹‘𝑘) = -∞) → 𝐷 ∈ ℝ) |
| 50 | 46, 49 | ltnled 11408 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) ∧ (𝐹‘𝑘) = -∞) → ((abs‘(-∞
− 𝐴)) < 𝐷 ↔ ¬ 𝐷 ≤ (abs‘(-∞ − 𝐴)))) |
| 51 | 34, 50 | mpbid 232 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) ∧ (𝐹‘𝑘) = -∞) → ¬ 𝐷 ≤ (abs‘(-∞ − 𝐴))) |
| 52 | 28, 51 | pm2.65da 817 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) → ¬ (𝐹‘𝑘) = -∞) |
| 53 | 52 | 3adant2 1132 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍 ∧ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) → ¬ (𝐹‘𝑘) = -∞) |
| 54 | 53 | neqned 2947 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍 ∧ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) → (𝐹‘𝑘) ≠ -∞) |
| 55 | | simpll 767 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝐹‘𝑘) ∈ ℂ) ∧ (𝐹‘𝑘) = +∞) → 𝜑) |
| 56 | | simpr 484 |
. . . . . . . . . . . . . 14
⊢ (((𝐹‘𝑘) ∈ ℂ ∧ (𝐹‘𝑘) = +∞) → (𝐹‘𝑘) = +∞) |
| 57 | | simpl 482 |
. . . . . . . . . . . . . 14
⊢ (((𝐹‘𝑘) ∈ ℂ ∧ (𝐹‘𝑘) = +∞) → (𝐹‘𝑘) ∈ ℂ) |
| 58 | 56, 57 | eqeltrrd 2842 |
. . . . . . . . . . . . 13
⊢ (((𝐹‘𝑘) ∈ ℂ ∧ (𝐹‘𝑘) = +∞) → +∞ ∈
ℂ) |
| 59 | 58 | adantll 714 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝐹‘𝑘) ∈ ℂ) ∧ (𝐹‘𝑘) = +∞) → +∞ ∈
ℂ) |
| 60 | | climxrrelem.p |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ +∞ ∈ ℂ)
→ 𝐷 ≤
(abs‘(+∞ − 𝐴))) |
| 61 | 55, 59, 60 | syl2anc 584 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝐹‘𝑘) ∈ ℂ) ∧ (𝐹‘𝑘) = +∞) → 𝐷 ≤ (abs‘(+∞ − 𝐴))) |
| 62 | 61 | adantlrr 721 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) ∧ (𝐹‘𝑘) = +∞) → 𝐷 ≤ (abs‘(+∞ − 𝐴))) |
| 63 | | fvoveq1 7454 |
. . . . . . . . . . . . . . 15
⊢ ((𝐹‘𝑘) = +∞ → (abs‘((𝐹‘𝑘) − 𝐴)) = (abs‘(+∞ − 𝐴))) |
| 64 | 63 | adantl 481 |
. . . . . . . . . . . . . 14
⊢
(((abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷 ∧ (𝐹‘𝑘) = +∞) → (abs‘((𝐹‘𝑘) − 𝐴)) = (abs‘(+∞ − 𝐴))) |
| 65 | | simpl 482 |
. . . . . . . . . . . . . 14
⊢
(((abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷 ∧ (𝐹‘𝑘) = +∞) → (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷) |
| 66 | 64, 65 | eqbrtrrd 5167 |
. . . . . . . . . . . . 13
⊢
(((abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷 ∧ (𝐹‘𝑘) = +∞) → (abs‘(+∞
− 𝐴)) < 𝐷) |
| 67 | 66 | adantll 714 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷) ∧ (𝐹‘𝑘) = +∞) → (abs‘(+∞
− 𝐴)) < 𝐷) |
| 68 | 67 | adantlrl 720 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) ∧ (𝐹‘𝑘) = +∞) → (abs‘(+∞
− 𝐴)) < 𝐷) |
| 69 | 42 | ad2antrr 726 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (𝐹‘𝑘) ∈ ℂ) ∧ (𝐹‘𝑘) = +∞) → 𝐴 ∈ ℂ) |
| 70 | 59, 69 | subcld 11620 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝐹‘𝑘) ∈ ℂ) ∧ (𝐹‘𝑘) = +∞) → (+∞ − 𝐴) ∈
ℂ) |
| 71 | 70 | abscld 15475 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝐹‘𝑘) ∈ ℂ) ∧ (𝐹‘𝑘) = +∞) → (abs‘(+∞
− 𝐴)) ∈
ℝ) |
| 72 | 71 | adantlrr 721 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) ∧ (𝐹‘𝑘) = +∞) → (abs‘(+∞
− 𝐴)) ∈
ℝ) |
| 73 | 48 | ad2antrr 726 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) ∧ (𝐹‘𝑘) = +∞) → 𝐷 ∈ ℝ) |
| 74 | 72, 73 | ltnled 11408 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) ∧ (𝐹‘𝑘) = +∞) → ((abs‘(+∞
− 𝐴)) < 𝐷 ↔ ¬ 𝐷 ≤ (abs‘(+∞ − 𝐴)))) |
| 75 | 68, 74 | mpbid 232 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) ∧ (𝐹‘𝑘) = +∞) → ¬ 𝐷 ≤ (abs‘(+∞ − 𝐴))) |
| 76 | 62, 75 | pm2.65da 817 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) → ¬ (𝐹‘𝑘) = +∞) |
| 77 | 76 | 3adant2 1132 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍 ∧ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) → ¬ (𝐹‘𝑘) = +∞) |
| 78 | 77 | neqned 2947 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍 ∧ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) → (𝐹‘𝑘) ≠ +∞) |
| 79 | 20, 54, 78 | xrred 45376 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍 ∧ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) → (𝐹‘𝑘) ∈ ℝ) |
| 80 | 14, 15, 18, 79 | syl3anc 1373 |
. . . . 5
⊢ (((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷))) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → (𝐹‘𝑘) ∈ ℝ) |
| 81 | 13, 80 | jca 511 |
. . . 4
⊢ (((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷))) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → (𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ ℝ)) |
| 82 | 5, 81 | ralrimia 3258 |
. . 3
⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷))) → ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ ℝ)) |
| 83 | 9 | ffund 6740 |
. . . . 5
⊢ (𝜑 → Fun 𝐹) |
| 84 | | ffvresb 7145 |
. . . . 5
⊢ (Fun
𝐹 → ((𝐹 ↾
(ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶ℝ ↔
∀𝑘 ∈
(ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ ℝ))) |
| 85 | 83, 84 | syl 17 |
. . . 4
⊢ (𝜑 → ((𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶ℝ ↔
∀𝑘 ∈
(ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ ℝ))) |
| 86 | 85 | adantr 480 |
. . 3
⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷))) → ((𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶ℝ ↔
∀𝑘 ∈
(ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ ℝ))) |
| 87 | 82, 86 | mpbird 257 |
. 2
⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷))) → (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶ℝ) |
| 88 | | breq2 5147 |
. . . . . 6
⊢ (𝑥 = 𝐷 → ((abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥 ↔ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) |
| 89 | 88 | anbi2d 630 |
. . . . 5
⊢ (𝑥 = 𝐷 → (((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥) ↔ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷))) |
| 90 | 89 | rexralbidv 3223 |
. . . 4
⊢ (𝑥 = 𝐷 → (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥) ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷))) |
| 91 | 41 | simprd 495 |
. . . 4
⊢ (𝜑 → ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ ℤ ∀𝑘 ∈
(ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥)) |
| 92 | 90, 91, 47 | rspcdva 3623 |
. . 3
⊢ (𝜑 → ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) |
| 93 | | climxrrelem.m |
. . . 4
⊢ (𝜑 → 𝑀 ∈ ℤ) |
| 94 | 6 | rexuz3 15387 |
. . . 4
⊢ (𝑀 ∈ ℤ →
(∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷) ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷))) |
| 95 | 93, 94 | syl 17 |
. . 3
⊢ (𝜑 → (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷) ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷))) |
| 96 | 92, 95 | mpbird 257 |
. 2
⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) |
| 97 | 87, 96 | reximddv 3171 |
1
⊢ (𝜑 → ∃𝑗 ∈ 𝑍 (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶ℝ) |